What is exactly called a Henkin model? Is this notion tied to 2nd order logic? How it differs from other non-Henkin's models?

  • $\begingroup$ encyclopediaofmath.org/index.php/Henkin_construction $\endgroup$ – realdonaldtrump May 27 '18 at 18:30
  • $\begingroup$ @realdonaldtrump I suspect that they're actually talking about Henkin vs. standard semantics. $\endgroup$ – Noah Schweber May 27 '18 at 18:32
  • $\begingroup$ @NoahSchweber You are right! $\endgroup$ – user122424 May 27 '18 at 18:34
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    $\begingroup$ @user122424 When you ask a question of the form "what is X?", there's always room for more context. Where did you encounter concept X? Was a definition given there? Or other references? If so, what did you find challenging about understanding concept X? This extra context will help you get more informative answers (like Noah's below), instead of just "here, read a definition" (like realdonaldtrump's comment above). It will also help avoid confusions like the one that occurred here, since many terms in mathematics refer to multiple things. $\endgroup$ – Alex Kruckman May 27 '18 at 19:23

There are two things one could plausibly mean by the phrase "Henkin model." The first is the sort of structure that emerges during the usual proof of Godel's Completeness Theorem (no, that's not a typo): a structure whose terms in a language mod provable equality in some theory. This is also called a "term model."

I suspect, however, that you're looking for the other meaning, based on your comment about second-order logic. The issue here is that there are two different ways we can approach second-order logic semantically:

  • We can think of a second-order structure as a first-order structure $\mathfrak{M}=(M; \mathcal{I})$ - with underlying set $M$, and "interpretation" $\mathcal{I}$ - together with a family $\mathbb{S}$ of subsets of/relations on/functions on the domain $M$ which our second-order quantifiers range over; and we demand that $\mathbb{S}$ be closed under "basic operations" so that things aren't silly (I'll leave this vague for now). This idea gives us two degrees of freedom in constructing a second-order structure: the choice of first-order part, and the separate (constrained, but not totally determined) choice of second-order part. This is the Henkin semantics, and a structure of the type above is a Henkin model.

  • Alternately, we can demand that second-order quantifiers really range over all the subsets of/relations on/functions on the domain $M$, so that the second-order part of a structure is completely determined by the first-order part (hence, even though our structures are more complicated, we still only have one degree of freedom in describing them). This is the standard semantics, and is the more commonly used semantics to my understanding (outside of computer science).

These two notions of semantics behave fundamentally differently. Henkin semantics is essentially first-order logic all over again, whereas the standard semantics is fundamentally different (and it's the standard semantics that people are referring to when they say e.g. "second-order logic is not compact" or similar). The standard semantics runs into serious set-theoretic difficulties right out the gate, which are detailed at many locations on this site and mathoverflow (see e.g. here, here, or here); on the other hand, we can also interpret this as saying that the standard semantics captures something fundamentally new, whereas the Henkin semantics is just clever language.

  • $\begingroup$ Note, incidentally, that every "standard model" is a Henkin model; the point is that the Henkin semantics allows for more models, which aren't considered legit by the standard semantics. E.g. if you adopt the standard semantics, you could think of "Henkin model" as being a polite way to say "pseudo-model." $\endgroup$ – Noah Schweber May 27 '18 at 18:39

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